NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 47 - 54 of 101
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Question 47
Appeared in Year: 2009
Write in Short
Short Answer▾Let
Write down the residues of at each of its poles
Question 48
Appeared in Year: 2007
Describe in Detail
Essay▾Let C denote the Boundary of the square whose sides are given by the lines . Assume that C is described in the position sense, i.e.. anticlockwise. Evaluate
Explanation
To evaluate
Take
Here is a pole of order 1 and lies within the square.
These are simple poles which lie outside the square
So they will not contribute towards the residue of
By Cauchy residue theorem
Question 49
Appeared in Year: 2014
Describe in Detail
Essay▾Let be differentiable at and let . Evaluate
Explanation
Now
So this is form (Exponential form)
Now using the result
Or if
Then
… eq. (1)
form
From (1)
… (1 more words) …
Question 50
Appeared in Year: 2009
Describe in Detail
Essay▾Let C be the contour consisting of the lines , described counterclockwise in the plane. Compute
Explanation
is the contour shown in the figure
To evaluate
Now is a pole of order 1 (simple pole)
Also lies within C
=
By Cauchy residue theorem
Question 51
Appeared in Year: 2017
Describe in Detail
Essay▾Let Write down the Taylor Expansion of f in the neighbourhood of .
Explanation
Here
Put
This is the required Taylor expansion of f in the neighbourhood of
Question 52
Appeared in Year: 2015
Describe in Detail
Essay▾In each of the following cases, state whether the given series is absolutely convergent, conditionally convergent or divergent
a)
b)
c)
Explanation
Option (a) The given series is
Where
Now comparing it with
1)
2)
3)
By Leibnitz test (alternating series test)
The given series is convergent
Which is finite & non – zero
converge or diverge together
But is divergent by p – test
is divergent (By comparison test)
is not absolutely convergent
is conditionally convergent
Option (b) The given series i…
… (44 more words) …
Question 53
Appeared in Year: 2006
Question
MCQ▾Pick out series which are absolutely convergent.
Choices
Choice (4) | Response | |
---|---|---|
a. | ||
b. | ||
c. | ||
d. | Both a. and b. are correct |
Question 54
Appeared in Year: 2005
Describe in Detail
Essay▾What are the value of for which the following series is convergent?
Explanation
Let the omen series be denoted by … eq. (1)
Where
Since for absolute convergence, we take
Now is convergent if … eq. (2)
And divergent if (By p-test)
Also eq. (1) is convergent for by Leibnitz test
is convergent by Leibnitz test for … eq. (3)
From eq. (2) & (3)
We get eq. (1) is convergent
… (3 more words) …